I am struggling to answer the Art of Electronics (3rd Ed.) exercise 2.2. I have found the answer given in the text is wrong and should be 76us (found an errata page on the web). But that's not the point...

What we have is a cap charged at 4.4V, suddenly having it's positive side switched to 0V. This means it suddenly has a -4.4V charge as seen on its other pin. Now it looks like a 10n charging through a 10k from 5V (tau=RC=100us) and the circuit output will switch when it reaches 0.6V (it's tied to an npn base). Simplified circuit below:

My immediate thought was "easy, I'll just use the capacitor charge equation". It turns out this is wrong. I did manage to find the right answer, but it was through guesswork and as I don't understand it it is pointless.

To get the right answer I had to use the capacitor DIScharge equation V=V0.ln(e^-t/RC). To confuse things even more I used an initial voltage of 9.4V and a final voltage of 4.4V (the initial and final voltage across the 10k). I feel like I am borderline understanding something but I just can't seem to make any progress!

Thanks for any help.

G